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According to Eqs. (
2.139
)-(
2.142
), if we let
u
i
1
j
=
1
u
ij
,
u
i
2
j
=
1
u
ij
,...,
u
ip
j
=
1
u
ij
w
(
i
)
=
,
1
≤
i
≤
c
(2.143)
then the prototypical IFSs
V
= {
V
1
,
V
2
,...,
V
c
}
of the IFCM algorithm can be
computed as:
w
(
i
)
)
V
i
=
f
(
A
,
⎧
⎨
⎫
⎬
⎭
,
x
s
,
p
p
w
(
i
)
j
w
(
i
)
j
=
μ
A
j
(
x
s
),
v
A
j
(
x
s
)
1
≤
s
≤
n
1
≤
i
≤
c
(2.144)
⎩
j
=
1
j
=
1
Since Eqs. (
2.138
) and (
2.144
) are computationally interdependent, we exploit an
iterative procedure similar to the fuzzy C-means to solve these equations. The steps
are as follows:
Algorithm 2.7
(IFCM algorithm)
Step 1
Initialize the seed
V
(
0
)
,let
k
=
0, and set
ε>
0.
Step 2
Calculate
U
(
k
)
=
(
u
ij
(
k
))
c
×
p
, where
(1) If for all
j
,
r
,
d
1
(
A
j
,
V
r
(
k
)) >
0, then
1
r
=
1
d
NE
(
A
j
,
V
i
(
k
))
u
ij
(
k
)
=
m
−
1
,
1
≤
i
≤
c
,
1
≤
j
≤
p
(2.145)
2
d
NE
(
A
j
,
V
r
(
k
))
(2) If there exist
j
and
r
such that
d
NE
(
A
j
,
V
r
(
k
))
=
0, then let
u
rj
(
k
)
=
1
and
u
ij
(
k
)
=
0, for all
i
=
r
.
Step 3
Calculate
V
(
k
+
1
)
= {
V
1
(
k
+
1
),
V
2
(
k
+
1
),...,
V
c
(
k
+
1
)
}
, where
w
(
i
)
(
V
i
(
k
+
1
)
=
f
(
A
,
k
+
1
)),
1
≤
i
≤
c
(2.146)
where
u
i
1
(
)
j
=
1
u
ij
(
k
)
j
=
1
u
ij
(
u
i
2
(
k
)
j
=
1
u
ij
(
u
ip
(
k
w
(
i
)
(
k
+
1
)
=
)
,
)
,...,
,
1
≤
i
≤
c
k
k
k
)
(2.147)
Step 4
If
i
=
1
d
1
(
V
i
(
k
),
V
i
(
k
+
1
))
<ε
, then end the algorithm; otherwise, let
k
:=
c
k
+
1, and return to Step 2.
For cases where the collected data are expressed as IVIFSs, Xu and Wu (2010)
extended Algorithm 2.7 to the interval-valued intuitionistic fuzzy C-means (IIFCM)
algorithm. We take the basic distance measure (
2.134
) as the proximity function of
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